Large deviations of a forced velocity jump process with a Hamilton-Jacobi approach

TL;DR

This paper investigates the large deviations of a forced velocity jump process using a Hamilton-Jacobi framework, revealing convergence of a potential to a solution with possible singularities, as a step towards understanding spreading phenomena.

Contribution

It introduces a Hamilton-Jacobi approach to analyze large deviations in a velocity jump process, including handling singular Hamiltonians.

Findings

Convergence of the potential to a Hamilton-Jacobi solution.

Identification of C1 singularities in the Hamiltonian.

Framework for future spreading studies in realistic processes.

Abstract

We study the dispersion of a particle whose motion dynamics can be described by a forced velocity jump process. To investigate large deviations results, we study the Chapman-Kolmogorov equation of this process in the hyperbolic scaling (t,x,v) -> (t/epsilon,x/epsilon,v) and then, perform a Hopf-Cole transform which gives us a kinetic equation on a potential. We prove the convergence of this potential to the solution of a Hamilton-Jacobi equation. The hamiltonian can have a C1 singularity, as was previously observed in this kind of studies. This is a preliminary work before studying spreading results for more realistic processes.